SP#7: Unit Q Concept 2 and Unit O Concept 4: Solving Clues using Identities and SOHCAHTOA

“This SP7 was made in collaboration with Preston Phan.  Please visit the other awesome posts on their blog by going here”

Solving Using Identities

To solve using SOHCAHTOA 
Select here

There are other possible ways of solving these types of problems since they are so broad as to what path we are allowed to take. What matters is our ending answer. We must take in consideration that when we are solving using identities we can always start off in a different step it does not have to be in the exact order that we have done it in but it can be solving something else first. The signs are also very important since it classifies where in the graph this triangle lies on. Also when we are using sohcahtoa the angle that we are using is the angle that is on the origin this is very important since it will define what our answers will be.

I/D3: Unit Q Concept 1: Pythagorean Identities

1. Inquiry Activity Summary 

sin^2x + cos^2x= 1

Where does sin^2x+cos^2x=1 come from? 

• It comes from the Pythagorean theorem. The Pythagorean theorem is an identity, which is a proven factor or formula that is always true. (Kirch) The Pythagorean theorem is an identity for that same reason since it has been proven and it is always true. 

What is the Pythagorean theorem using x,y, and r? 

• We know that the Pythagorean Theorem is a^2+b^2=c^2.
• When we plot it in coordinate plane using the unit circle we get x^2+y^2=r^2. "x" being the horizontal leg, "y" being the vertical leg of the triangle and "r" being the hypotenuse which is equal to one since we are dealing with the unit circle.

Pythagorean theorem equal to 1

• In order to have the Pythagorean Theorem equal to 1 we divide by r^2 in order to give us the product of 1. The equation would now be x^2/r^2 + y^2/r^2 =1 which can be simplified to a nicer form of -
  

The ratio of cosine on the unit circle is x/r 

The ratio for sine on the unit circle is y/r. 

What do you notice? 

• We notice that what is being squared in our previous equation are the values that we have just found out. They are the values of what sin stands for and what cos stand for that are being squared and the value of r is one because that is the length of the hypotenuse always one on the unit circle. 

We can conclude:

• We can conclude that sin^2x + cos^2x= 1 is an identity as well because it has been proven and is always true because it follows the Pythagorean theorem method.

How do we know it is true? 

• We plug it in which demonstrates that the equation is true, it is valid since it is equal to 1 just like the equation had stated it would be. 
• making this an identity since it has been proven and it will always be true. 

Derive to identify with secant and Tangent

What do we know?

sin^2x + cos^2x =1

• we simplify this in a way that will give us both secant and tangent in only one step
• in order to get this we will divide everything by cos^2x 
• Which will end up giving us sin^2x/cos^2x + cos^2x/ cos^2x= 1/cos^2  
• which can be simplified to tan^2x + 1 = sec^2x 

Derive to Identify Cosecant and Cotangent 

We know sin^2x + cos^2x = 1 is where we are starting from 

•  we first look at what is being asked for us so we can solve in just one step
• in order to get both cosecant and cotangent into the equation we must divide by sin^2x
• this will end up giving us 1+ cot^2x= csc^2x which has all that is being asked to have 

2. Inquiry Activity Reflection 

1. The connections that i see between units N,O,P so far are that they are all coming from the unit circle, all triangles eventually end up connecting to the unit circle and a right triangle. Also how in units N,O, and P there are connection between knowing the trig ratios, because they will come in handy the further along we get into the unit. Knowing the trig functions help us better understand what is going on and where everything is coming from.

2. If i had to describe trigonometry in three words it would be complex, rigorous, and interesting. This is so complex because there are many parts to it and it test what we have learned previously. It is rigorous because not only is it not learned in a short amount of times but we have to understand the connection between everything in order to fully understand the concept. Interesting, because once a light bulb goes off in your head the world finally makes sense and  you feel like you can conquer the world.

WPP#13-14: Unit P Concepts 6-7: Solving law of Sin and Cosine word Problems: The Skydiving Pizzeria Evening

“This WPP13-14 was made in collaboration with Preston.  Please visit the other awesome posts on their blog by going here”

a) One day Nancy decided that she was ready for some excitement and adrenaline in her life. She decided to go skydiving but to add a little more adrenaline she wanted her friend Alejandra to be waiting for her on the west at the bottom of her fall. On the east is a pizzeria. When Alejandra looks up to see her friend Nancy she has to look N 39 E. The people inside the pizzeria spot Nancy as well and have to look N 47 W. The distance between Alejandra and the Pizzeria is 60 ft. (assuming that the airplane is exactly half way between Alejandra and the Pizzeria.) What is the distance that Nancy has to travel in order to reach the ground? 


b) Alejandra and Nancy have eaten at the Pizzeria by now and are ready to go their separate ways. Alejandra went on a baring of 150 degrees. While Nancy went on a baring of 245 degrees. Alejandra goes 25 mph for 20 minutes and Nancy walks 13 mph for 30 minutes. What is the distance between Alejandra and Nancy? 

Solutions

a)

b)

BQ#1: Unit P Concepts 1-2, 4: How to Derive the Law of Sines and How to find the area of an Oblique triangle

1. Law of Sines

We need the law of sines because no all triangles will end up being right triangles and we need to know how to solve for all the missing parts of the triangle. If we thing of what is around us we will not be able to know exactly what everything is unless we know some kind of rule or equation to help us figure this out. 


4. Area of an Oblique triangle
This equation is related to the equation of finding the area of a triangle because that is what we are using. The only thing that we are changing is how the height is being represented. instead of writing h we are using asinC and that is still equivalent to the height.

WPP#12: Unit O Concept 10: How to Solve Depression and Elevation Word Problems

http://us.cdn3.123rf.com/168nwm/argument/argument0901/argument090100156/4205169-young-woman-on-the-edge-of-petersburg-s-roof.jpg

Tiana decides to climb on her roof because she was curios to see what is on top of the building from her house that is 309 ft away. She tries to see what is up there with an elevation of 33 degrees, from her eyes to the top of the building. She then started to get curious as to what is that supports that glass building and keeps it strong, and looks down. The height between the bottom of the building and where her eyes are located is 229 ft.

a) What is the height between her eyes to the top of the building?

b) What is the angle of depression? 

Solution:

I/D2: Unit O- How to Derive Special Right Triangles

Inquiry Activity 
1. Deriving a 45-45-90 triangle
 Click Here

2. Deriving a 30-60-90 triangle
Click Here

3. The reason why both of these the 30-6-90 triangle and the 45-45-90 triangle have variables of "n" and not just the numbers because there is a relationship between as to why these numbers are there and they end up being a patter for all triangles of that kind. The variables make it a rule and make it true for all the special right triangles.

Inquiry Activity Reflection

1.Something I never noticed about special right triangles is... that all the sides have a pattern that is common between that type of right triangle. For example, In a 45-45-90 triangle we know that "a" and "b" will be the same since they have the same angle value. The hypotenuse however has an extra radical 2 being multiplied to the value of both "a" and "b". For a 30-60-90 triangle we see that the length of the shortest side which could be classified as "a" has a value of n. The value of the medium size side is n radical 3, and the side of the hypotenuse is 2n. Which makes since we see that the hypotenuse is suppose to be greater since it is the longest side and it is in the equation.

2. Being able to derive these patterns myself aid in my learning because... it helps me understand how everything connects and allows me to fully understand how to solve each and every problem. This can be very important to me in the long run because if i ever forget the rules i can easily derive them using the Pythagorean theorem and will help me connect everything else to the unit circle and not just with a radius (hypotenuse) of 1 but of any number and can solve it.